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Optimization problem in computer science
In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability
Lattice_problem
Important problem in lattice theory
congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed
Congruence_lattice_problem
Cryptographic primitives that involve lattices
certain average-case lattice problem, known as short integer solutions (SIS), is at least as hard to solve as a worst-case lattice problem. She then showed
Lattice-based_cryptography
finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of
Finite lattice representation problem
Finite_lattice_representation_problem
Geometrical structure
density around 63.5%. A lattice arrangement (commonly called a regular arrangement) is one in which the points of the lattice form a very symmetric pattern
Sphere_packing
Periodic set of points
Coordinate-wise addition or subtraction of two points in the lattice produces another lattice point. The lattice points are all separated by some minimum distance
Lattice_(group)
Computational problem used in cryptography
solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began
Short integer solution problem
Short_integer_solution_problem
24-dimensional repeating pattern of points
Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, E24. It is one of the best models for the kissing number problem. It
Leech_lattice
Method to determine the electronic structure of strongly correlated materials
consists in mapping a many-body lattice problem to a many-body local problem, called an impurity model. While the lattice problem is in general intractable
Dynamical_mean-field_theory
Mathematical problem in cryptography
learning problem. Regev showed that the LWE problem is as hard to solve as several worst-case lattice problems. Subsequently, the LWE problem has been
Learning_with_errors
Quantum chromodynamics on a lattice
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge
Lattice_QCD
How many integer lattice points there are in a circle
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and
Gauss_circle_problem
Lattice in 8-dimensional space with special properties
mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name
E8_lattice
Hypothesis in computational complexity theory
assumptions used in cryptography (including RSA, discrete log, and some lattice problems) can be based on worst-case assumptions via worst-case-to-average-case
Computational hardness assumption
Computational_hardness_assumption
Quantum-safe key encapsulation mechanism
asymmetric cryptosystem uses a variant of the learning with errors lattice problem as its basic trapdoor function. It won the NIST competition for the
ML-KEM
Partially ordered set in which all subsets have both a supremum and infimum
complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice satisfies
Complete_lattice
Mathematical operation
exponential in the dimension of the lattice. Finding a reduced lattice basis is also closely related to the problem in crystallography of finding a unique
Lattice_reduction
Model in Quantum Physics
mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the
Particle in a one-dimensional lattice
Particle_in_a_one-dimensional_lattice
Problems which attempt to find the most efficient way to pack objects into containers
nine possible definable packings. The 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective
Packing_problems
Problem in computer science
perform for this task, as it can be applied to break lattice-based cryptography. The hidden shift problem states: Given an oracle O {\displaystyle O} that
Hidden_shift_problem
Construction analogous to that of a dual vector space
theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a
Dual_lattice
Computational problem possibly useful for post-quantum cryptography
errors problem is the fact that the solution to the RLWE problem can be used to solve a version of the shortest vector problem (SVP) in a lattice (a polynomial-time
Ring_learning_with_errors
Mathematical object
assumption that the shortest vector problem (SVP) is hard in these ideal lattices. In general terms, ideal lattices are lattices corresponding to ideals in rings
Ideal_lattice
Geometry and crystallography point array
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of
Bravais_lattice
Project by NIST to standardize post-quantum cryptography
various intellectual property concerns were voiced, notably surrounding lattice-based schemes such as Kyber and NewHope. NIST holds signed statements from
NIST Post-Quantum Cryptography Standardization
NIST_Post-Quantum_Cryptography_Standardization
scenario the protein folding problem is NP-complete. Different versions of lattice proteins may adopt different types of lattice (typically square and triangular
Lattice_protein
Mathematical concept
free lattice is the free object corresponding to a lattice. As free objects, they have the universal property. Because the concept of a lattice can be
Free_lattice
Optimization problem in computer science
Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding
Nearest_neighbor_search
Algorithm in computational number theory
The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Lattice-based cryptosystem
Goldreich–Goldwasser–Halevi (GGH) lattice-based cryptosystem is a broken asymmetric cryptosystem based on lattices. There is also a GGH signature scheme
GGH_encryption_scheme
Algorithm to be run on quantum computers
generalization of the previously mentioned problems, as well as graph isomorphism and certain lattice problems. Efficient quantum algorithms are known for
Quantum_algorithm
Very general problem in computer science
quantum algorithms for two major problems: the graph isomorphism problem and certain shortest vector problems (SVPs) in lattices. More precisely, an efficient
Hidden_subgroup_problem
Equivalence relation in algebra
category § Definition for details. Chinese remainder theorem Congruence lattice problem Table of congruences Since a′−1 = a′−1 * a * a−1 ~ a′−1 * a′ * a−1
Congruence_relation
Mathematical puzzle
(1967). "536 Puzzles And Curious Problems". p. 376. Klamkin, M. S. (1955-02-01). "Polygonal Path Covering a Square Lattice (E1123)". The American Mathematical
Nine_dots_puzzle
Geometric concept
for each individual sphere as the number of spheres it touches. For a lattice packing, the kissing number is the same for every sphere; but for an arbitrary
Kissing_number
Hungarian-American computer scientist
Ajtai and Dwork devised in 1997 a lattice-based public-key cryptosystem; Ajtai has done extensive work on lattice problems. For his numerous contributions
Miklós_Ajtai
Mathematical model of ferromagnetism in statistical mechanics
of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing
Ising_model
one-dimensional (1D) lattice problem. In 1944 Onsager was able to get an exact solution to a two-dimensional (2D) lattice problem at the critical density
Lattice density functional theory
Lattice_density_functional_theory
Korkine–Zolotarev (KZ) lattice basis reduction algorithm or Hermite–Korkine–Zolotarev (HKZ) algorithm is a lattice reduction algorithm. For lattices in R n {\displaystyle
Korkine–Zolotarev lattice basis reduction algorithm
Korkine–Zolotarev_lattice_basis_reduction_algorithm
1979 conjecture in combinatorics
at most half the lattice, with equality only if the lattice is a Boolean lattice. As Abe (2000) shows, this statement about lattices is equivalent to
Union-closed_sets_conjecture
Topics referred to by the same term
COIN-OR Linear Program Solver Communication Linking Protocol Congruence lattice problem Constraint Logic Programming Constraint logic programming (Real) Control
CLP
Summatory function of the divisor-counting function
this problem remains unsolved. Progress has been slow. Many of the same methods work for this problem and for Gauss's circle problem, another lattice-point
Divisor_summatory_function
Putting fermions on a lattice with chiral symmetry results in more fermions than expected
although the fermion doubling problem remains in arbitrary dimensions and even if interactions are included. Lattice field theory is usually carried
Fermion_doubling
Mathematical proof technique
terms of lattice points on hyperbolas in the first quadrant. The same process of finding smaller roots is used instead to find lower lattice points on
Vieta_jumping
Cryptography secured against quantum computers
Ring-LWE there is a security reduction to the shortest-vector problem (SVP) in a lattice as a lower bound on the security. The SVP is known to be NP-hard
Post-quantum_cryptography
Theory of the strong nuclear interactions
and antiquarks in a meson. However, the numerical sign problem makes it difficult to use lattice methods to study QCD at high density and low temperature
Quantum_chromodynamics
Geometry problem on grid points
William; Pach, János (2005). "Section 10.1: Packing lattice points in subspaces". Research Problems in Discrete Geometry. Springer, New York. pp. 417–421
No-three-in-line_problem
Bound lattice in which every element has a complement
the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every
Complemented_lattice
Dense arrangement of congruent spheres in an infinite, regular arrangement
stacked upon one another. The FCC lattice is also known to mathematicians as that generated by the A3 root system. The problem of close-packing of spheres was
Close-packing of equal spheres
Close-packing_of_equal_spheres
Seven mathematical problems with a US$1 million prize for each solution
to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch
Millennium_Prize_Problems
Decision problem pertaining to equivalence of expressions
Heyting algebra). The word problem on free lattices and more generally free bounded lattices has a decidable solution. Bounded lattices are algebraic structures
Word_problem_(mathematics)
Algebra whose elements are stable matchings
a lattice of stable matchings is a distributive lattice whose elements are all the solutions to a given instance of the stable matching problem. These
Lattice_of_stable_matchings
Result of partitioning the elements of an algebraic structure using a congruence relation
associated with congruence identities. Quotient ring Congruence lattice problem Lattice of subgroups A. G. Kurosh, Lectures on General Algebra, Translated
Quotient_(universal_algebra)
Problem in applied mathematics
lattice QCD to predict the phases and properties of quark matter. (In lattice field theory, the problem is also known as the complex action problem.)
Numerical_sign_problem
Class of computational fluid dynamics methods
The lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy-Pomeau-Pazzis and Frisch-Hasslacher-Pomeau models), is
Lattice_Boltzmann_methods
Quasiparticle of mechanical vibrations
Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modeling phonons. (For other common lattices, see crystal
Phonon
Farrell–Jones conjecture Finite lattice representation problem: is every finite lattice isomorphic to the congruence lattice of some finite algebra? Goncharov
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Norwegian-American physical chemist and theoretical physicist (1903-1976)
Norway portal Biography portal Lattice density functional theory (1944) solution to a two-dimensional (2D) lattice problem Lars Onsager at the Mathematics
Lars_Onsager
Theory of quantum gauge fields on a lattice
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important
Lattice_gauge_theory
Matrix form in linear algebra
Hermite normal form must be used. Given two bases for a lattice, A and A', the equivalence problem is to decide if L A = L A ′ . {\displaystyle L_{A}=L_{A'}
Hermite_normal_form
Ordered arrangement of atoms, ions, or molecules in a crystalline material
the Bravais lattice. The lengths of principal axes/edges, of the unit cell and angles between them are lattice constants, also called lattice parameters
Crystal_structure
Regular infinite tree structure used in statistical mechanics
Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors. The Bethe lattice was
Bethe_lattice
German physicist (1900–1998)
the Hopfield network (1982). Lattice density functional theory (1925) solution to the one-dimensional (1D) lattice problem Stutz, Conley; Williams, Beverly
Ernst_Ising
has been applied recently in solving several lattice theory problems, such as the congruence lattice problem. Denote by [ X ] < ω {\displaystyle [X]^{<\omega
Kuratowski's_free_set_theorem
Apparent paradox in chaos theory
In physics, the Fermi–Pasta–Ulam–Tsingou (FPUT) problem or formerly the Fermi–Pasta–Ulam problem was the apparent paradox in chaos theory that many complicated
Fermi–Pasta–Ulam–Tsingou problem
Fermi–Pasta–Ulam–Tsingou_problem
Type of cellular automaton
Lattice gas automata (LGCA), or lattice gas cellular automata, are a type of cellular automaton used to simulate fluid flows, pioneered by Hardy–Pomeau–de
Lattice_gas_automaton
List of unsolved computational problems
shortest vector of a lattice be computed in polynomial time on a classical or quantum computer? Can the graph isomorphism problem be solved in polynomial
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Two-port electrical wave filter
A symmetrical lattice is a two-port electrical wave filter in which diagonally-crossed shunt elements are present – a configuration which sets it apart
Lattice_network
American mathematician
Borel equivalence relations. With Dan Mauldin he solved the Steinhaus lattice problem. Jackson earned his PhD in 1983 at UCLA under the direction of Donald
Steve_Jackson_(mathematician)
Type of observation mast on warships
Lattice masts, or cage masts, or basket masts, are a type of observation mast common on United States Navy major warships in the early 20th century. They
Lattice_mast
Medical condition
Sometimes other retinal problems (such as tears, breaks, or holes) may be present along with lattice degeneration. However, these problems may also be distinct
Lattice_degeneration
Conceptual conflict between general relativity and quantum mechanics
In theoretical physics, the problem of time is a conceptual conflict between quantum mechanics and general relativity. Quantum mechanics regards the flow
Problem_of_time
Lattice in universal algebra
In logic and universal algebra, Post's lattice denotes the lattice of all clones on a two-element set {0, 1}, ordered by inclusion. It is named for Emil
Post's_lattice
Function used in computer cryptography
decoding of random linear codes, the hardness of certain lattice problems, and the subset sum problem (Naccache–Stern knapsack cryptosystem). There is an explicit
One-way_function
Sequence of moves on a lattice
Unsolved problem in mathematics Is there a formula or algorithm that can calculate the number of self-avoiding walks in any given lattice? More unsolved
Self-avoiding_walk
Mathematical logic concept
under set inclusion; this poset is a lattice. The theory of this lattice is known to be an undecidable problem. Similarly, the set of all computably
Computably_enumerable_set
Application of geometry in number theory
approximation, the problem of finding rational numbers that approximate an irrational quantity. Suppose that Γ {\displaystyle \Gamma } is a lattice in n {\displaystyle
Geometry_of_numbers
No-go theorem concerning chirality of regularized fermions
that the Standard Model cannot be put on a lattice. Common methods for overcoming the fermion doubling problem is to use modified fermion formulations such
Nielsen–Ninomiya_theorem
mathematical problems involving lattices. Unlike older lattice based cryptographic algorithms, the RLWE-KEX is provably reducible to a known hard problem in lattices
Ring learning with errors key exchange
Ring_learning_with_errors_key_exchange
Threshold of percolation theory models
occurs. The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites
Percolation_threshold
Numerical method used in computational fluid dynamics
The Vortex lattice method, (VLM), is a numerical method used in computational fluid dynamics, mainly in the early stages of aircraft design and in aerodynamic
Vortex_lattice_method
Probabilistic algorithms to simulate quantum many-body systems
designed for bosons that can simulate any complicated lattice Hamiltonian that does not have a sign problem. World-line quantum Monte Carlo Stochastic series
Quantum_Monte_Carlo
Approach to public-key cryptography
a central hardness assumption is the elliptic curve discrete logarithm problem (ECDLP): given a public base point P {\displaystyle P} and another point
Elliptic-curve_cryptography
Algorithm for public-key cryptography
numbers, the "factoring problem". Breaking RSA encryption is known as the RSA problem. Whether it is as difficult as the factoring problem is an open question
RSA_cryptosystem
Sequence of end-to-end vectors across points of a lattice
In combinatorics, a lattice path L in the d-dimensional integer lattice Z d {\displaystyle \mathbb {Z} ^{d}} of length k with steps in the set S,
Lattice_path
Natural number
and 24. The Leech lattice Λ24 is a 24-dimensional lattice through which 23 other positive definite even unimodular Niemeier lattices of rank 24 are built
23_(number)
Combinatorial sequence of numbers
{\displaystyle n} -element set, the number of elements in a free distributive lattice with n {\displaystyle n} generators, and one more than the number of abstract
Dedekind_number
Kind of microscopy
Lattice light-sheet microscopy is a modified version of light sheet fluorescence microscopy that increases image acquisition speed while decreasing damage
Lattice light-sheet microscopy
Lattice_light-sheet_microscopy
Digital signature resilient to quantum cryptography
based on hard problems in lattices are being created to replace the commonly used RSA and elliptic curve signatures. A subset of these lattice based scheme
Ring learning with errors signature
Ring_learning_with_errors_signature
following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics are theoretical, meaning
List of unsolved problems in physics
List_of_unsolved_problems_in_physics
Computational method
similar to Zassenhaus, except that the combinatorial problem is translated to a lattice problem that is then solved by LLL. In this approach, LLL is not
Factorization_of_polynomials
Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results, p. 12 O. R. Musin (2003). "The problem of the twenty-five spheres"
16-cell_honeycomb
Integer factorization algorithm
Lattice sieving is a technique for finding smooth values of a bivariate polynomial f ( a , b ) {\displaystyle f(a,b)} over a large region. It is almost
Lattice_sieving
the details of protein folding, it is still an NP-hard problem on both 2D and 3D square lattices. A Monte Carlo method, named FRESS, was developed and
Hydrophobic-polar protein folding model
Hydrophobic-polar_protein_folding_model
Sum of inverse squares of natural numbers
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed
Basel_problem
Mathematical question in algebraic geometry
particular type, arising from a lattice in Cg. In relatively concrete terms, it is being asked which lattices are the period lattices of compact Riemann surfaces
Schottky_problem
Pairing where no unchosen pair prefers each other over their choice
marriage problem can be given the structure of a finite distributive lattice, and this structure leads to efficient algorithms for several problems on stable
Stable_matching_problem
Problem in physics and quantum mechanics
theory Lattice gauge theory Matrix product state Neural network quantum states Numerical renormalization group Jenkins, Stephen. "The Many Body Problem and
Many-body_problem
Microscopic theory of superconductivity
pairs of electrons known as Cooper pairs. These pairs move through the lattice without resistance. The theory is named after John Bardeen, Leon Cooper
BCS_theory
In mathematics, the Tamari lattice is an algebraic structure that concisely represents some of the important logical and geometric properties of associativity
Tamari_lattice
LATTICE PROBLEM
LATTICE PROBLEM
Female
English
Pet form of English Matilda, MATTIE means "mighty in battle." Compare with masculine Mattie.
Female
English
Variant spelling of English Patty, PATTIE means "patrician; of noble birth."
Female
English
Pet form of French Charlotte, LOTTIE means "man."
Girl/Female
Latin
From Attica.
Girl/Female
Australian, French, Latin
Laurel
Girl/Female
American, Australian, British, English, Greek
Modern Blend of Catrina and Patrice
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddock.
Girl/Female
Latin
Joy. Popular medieval British form of the name Letitia.
Girl/Female
British, Christian, English, French, Latin
Joy; Popular Medieval Form of the Name Letitia; Gladness; Happiness
Male
English
Pet form of English Matthew, MATTIE means "gift of God." Compare with feminine Mattie.
Girl/Female
French Latin American
noble.
Male
French
Medieval French form of Latin Patricius, PATRICE means "patrician; of noble descent."
Female
French
French form of Latin Viatrix, BÉATRICE means "voyager (through life)."
Girl/Female
Latin
From Attica.
Boy/Male
Latin
Regal.
Female
English
Pet form of English Harriet, HATTIE means "little home-ruler."
Female
English
Pet form of Middle English Lettice, LETTIE means "happiness."
Surname or Lastname
English
English : variant spelling of Latimer.
Girl/Female
American, Australian, Christian, French, Greek, Hebrew
Weary; Tired; Delicate; A Combination of Leah and Beatrice; Voyager through Life
Female
English
Middle English form of Latin Lætitia, LETTICE means "happiness."
LATTICE PROBLEM
LATTICE PROBLEM
Boy/Male
Hindu, Indian, Kannada
Glorious One; Shining
Boy/Male
Russian Greek
Of the conquering people.
Boy/Male
Hindu
Male
German
Frisian pet form of Germanic names beginning with arn-, ANNE means "eagle." Compare with feminine Anne.
Boy/Male
Hindu, Indian
One with a Good Mind and who is Happy
Boy/Male
Hindu
Sage, Ray of light
Girl/Female
Australian, Dutch, French, Latin, Spanish
Worthy
Boy/Male
Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Tamil, Telugu
King of Poets; Name of Lord Ganesh
Girl/Female
Muslim
Purity, Modesty, Infallibility
Girl/Female
Polish Native American
Hope.
LATTICE PROBLEM
LATTICE PROBLEM
LATTICE PROBLEM
LATTICE PROBLEM
LATTICE PROBLEM
v. i.
To make a lattice of; as, to lattice timbers.
imp. & p. p.
of Lattice
n.
See Brattice, n.
a.
Shaped like a lattice; cancellate.
v. i.
To close, as an opening, with latticework; to furnish with a lattice; as, to lattice a window.
n.
The strong wooden lattice used to cover a hatch, admitting light and air; also, a movable Lattice used for the flooring of boats.
n.
A composite plant of the genus Lactuca (L. sativa), the leaves of which are used as salad. Plants of this genus yield a milky juice, from which lactucarium is obtained. The commonest wild lettuce of the United States is L. Canadensis.
v. t.
A lattice or grating.
n.
The representation of a piece of latticework used as a bearing, the bands being vertical and horizontal.
a.
Of or pertaining to Attica, in Greece, or to Athens, its principal city; marked by such qualities as were characteristic of the Athenians; classical; refined.
n.
Any work of wood or metal, made by crossing laths, or thin strips, and forming a network; as, the lattice of a window; -- called also latticework.
n.
The act or process of making a lattice of, or of fitting a lattice to.
a.
Latticed. See Lattice, n., 2.
n.
The language of the Lettic race, including Lettish, Lithuanian, and Old Prussian.
n.
A chantry chapel inclosed with lattice or screen work.
p. pr. & vb. n.
of Lattice
n.
A pointed wooden tool used in glazing leaden lattice.
a.
Formed in latticework; latticed.
a.
Of or pertaining to milk; procured from sour milk or whey; as, lactic acid; lactic fermentation, etc.
n.
Same as Lattice, n., 1.